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Chebyshevskii Sb., 2018, Volume 19, Issue 1, Pages 138–151 (Mi cheb627)  

Joint discrete universality for Lerch zeta-functions

A. Laurinčikas, A. Mincevič

Vilnius University

Abstract: After Voronin's work of 1975, it is known that some of zeta and $L$-functions are universal in the sense that their shifts approximate a wide class of analytic functions. Two cases of shifts, continuous and discrete, are considered.
The present paper is devoted to the universality of Lerch zeta-functions $L(\lambda, \alpha, s)$, $s= \sigma+it $, which are defined, for $ \sigma > 1$, by the Dirichlet series with terms $ e^{2 \pi i \lambda m}(m+ \alpha)^{-s} $ with parameters $\lambda \in \mathbb{R} $ and $\alpha$, $0 < \alpha \leqslant 1$, and by analytic continuation elsewhere. We obtain joint discrete universality theorems for Lerch zeta-functions. More precisely, a collection of analytic functions $ f_{1}(s), …, f_{r}(s) $ simultaneously is approximated by shifts $L(\lambda_{1},\alpha_{1},s+ikh),…, L(\lambda_{r},\alpha_{r},s+ikh)$, $k=0,1,2,…$, where $h>0$ is a fixed number. For this, the linear independence over the field of rational numbers for the set $\{ (\log (m+ \alpha_{j}): m \in \mathbb{N}_{0},\; j=1,…,r),\frac{2 \pi}{h} \}$ is required. For the proof, probabilistic limit theorems on the weak convergence of probability measures in the space of analytic function are applied.

Keywords: Lerch zeta-function, Mergelyan theorem, space of analytic functions, universality, weak convergence.

Funding Agency Grant Number
ESF - European Social Fund 09.3.3-LMT-K-712-01-0037
The research of the first author is funded by the European Social Fund according to the activity “Improvement of researchers” qualification by implementing world-class R&D projects' of Measure No. 09.3.3-LMT-K-712-01-0037.


DOI: https://doi.org/10.22405/2226-8383-2018-19-1-138-151

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UDC: 511.3

Citation: A. Laurinčikas, A. Mincevič, “Joint discrete universality for Lerch zeta-functions”, Chebyshevskii Sb., 19:1 (2018), 138–151

Citation in format AMSBIB
\Bibitem{LauMin18}
\by A.~Laurin{\v{c}}ikas, A.~Mincevi{\v{c}}
\paper Joint discrete universality for Lerch zeta-functions
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 1
\pages 138--151
\mathnet{http://mi.mathnet.ru/cheb627}
\crossref{https://doi.org/10.22405/2226-8383-2018-19-1-138-151}


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